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The sample covariance matrix has in the denominator rather than due to a variant of Bessel's correction: In short, the sample covariance relies on the difference between each observation and the sample mean, but the sample mean is slightly correlated with each observation since it is defined in terms of all observations.
The reason the sample covariance matrix has in the denominator rather than is essentially that the population mean is not known and is replaced by the sample mean ¯. If the population mean E ( X ) {\displaystyle \operatorname {E} (\mathbf {X} )} is known, the analogous unbiased estimate is given by
The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R p×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. [1]
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n − 1) / n; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n-1 instead of n, is called ...
In estimating the population variance from a sample when the population mean is unknown, the uncorrected sample variance is the mean of the squares of deviations of sample values from the sample mean (i.e., using a multiplicative factor 1/n). In this case, the sample variance is a biased estimator of the population variance. Multiplying the ...
The expected values needed in the covariance formula are estimated using the sample mean, e.g. = = and the covariance matrix is estimated by the sample covariance matrix (,) , where the angular brackets denote sample averaging as before except that the Bessel's correction should be made to avoid bias.
A similar important statistic in exploratory data analysis that is simply related to the order statistics is the sample interquartile range. The sample median may or may not be an order statistic, since there is a single middle value only when the number n of observations is odd.